Equilibrium in capacitated network models with queueing delays, queue-storage, blocking back and control

This paper considers a steady-state, link-based, fixed (or inelastic) demand equilibrium model with explicit link-exit capacities, explicit bottleneck or queuing delays and explicit bounds on queue storage capacities. The spatial queueing link model at the heart of this equilibrium model takes account of the space taken up by queues both when there is no blocking back and also when there is blocking back. The paper shows in theorem 1 that a feasible traffic assignment model has an equilibrium solution provided prices are used to impose capacity restrictions and utilises this result to show that there is an equilibrium with the spatial queueing model, provided queue-storage capacities are sufficiently large. Other results are obtained by changing the variables and sets in theorem 1 suitably. These results include: (1) existence of equilibrium results (in both a steady state and a dynamic context) which allow signal green-times to respond to prices and (2) an existence of equilibrium result which allow signal green-times to respond to spatial queues; provided each of these responses follows the P0 control policy described in Smith (1979a, 1987). These results show that under certain conditions the P0 control policy maximises network capacity. The operation of the spatial queueing link model is illustrated on a simple network. Finally the paper includes elastic demand; this is necessary for long-run evaluations. Each of the steady state equilibria whose existence is shown here may be thought of as a stationary solution to the dynamic assignment problem either with or without blocking back; they are quasi-dynamic equilibria. © 2013 The Authors. Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Delft University of Technology